1. **State the problem:** Divide the mixed number $1 \frac{4}{9}$ by the fraction $\frac{5}{9}$. 2. **Convert the mixed number to an improper fraction:** $$1 \frac{4}{9} = \frac{9}{9} + \frac{4}{9} = \frac{13}{9}$$ 3. **Recall the division rule for fractions:** To divide by a fraction, multiply by its reciprocal. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 4. **Apply the rule:** $$\frac{13}{9} \div \frac{5}{9} = \frac{13}{9} \times \frac{9}{5}$$ 5. **Simplify before multiplying:** The 9 in numerator and denominator cancels out. $$\frac{13}{\cancel{9}} \times \frac{\cancel{9}}{5} = \frac{13}{1} \times \frac{1}{5} = \frac{13}{5}$$ 6. **Convert the improper fraction back to a mixed number:** $$\frac{13}{5} = 2 \frac{3}{5}$$ **Final answer:** $$1 \frac{4}{9} \div \frac{5}{9} = 2 \frac{3}{5}$$